Chứng minh rằng \(a\left(b-c\right)\left(b+c-a\right)^2+c\left(a-b\right)\left(a+b-c\right)^2=b\left(a-c\right)\left(a+c-b\right)^2\)
Chứng minh rằng với mọi số thực a,b,c ta có:
\(a\left(b-c\right)\left(b+c-a\right)^2+c\left(a-b\right)\left(a+b-c\right)^2=b\left(a-c\right)\left(a+c-b\right)^2\\ \)
Chứng minh rằng nếu a,b,c khác nhau thì \(\frac{b-c}{\left(a-b\right)\left(a-c\right)}+\frac{c-a}{\left(b-c\right)\left(b-a\right)}+\frac{a-b}{\left(c-a\right)\left(c-b\right)}=\frac{2}{a-b}+\frac{2}{b-c}+\frac{2}{c-a}\)
Cho a,b,c khác nhau. Chứng minh rằng \(\frac{b-c}{\left(a-b\right)\left(a-c\right)}+\frac{c-a}{\left(b-c\right)\left(b-a\right)}+\frac{a-b}{\left(c-a\right)\left(c-b\right)}=\frac{2}{a-b}+\frac{2}{b-c}+\frac{2}{c-a}\)
Bài 1: Chứng minh rằng
\(a\left(b-c\right)\left(b+c-a\right)^2+c\left(a-b\right)\left(a+b-c\right)^2=b\left(a-c\right)\left(a+c-b\right)^2\)
Chứng minh rằng: \(\left(a^2+b^2+c^2\right)\left[\left(\dfrac{1}{\left(a-b\right)^2}+\dfrac{1}{\left(b-c\right)^2}+\dfrac{1}{\left(c-a\right)^2}\right)\right]\ge\dfrac{9}{2}\)
Với a, b, c là những số thực dương thỏa mãn \(\left(a+b\right)\left(b+c\right)\)\(\left(c+a\right)\)=1
Chứng minh rằng \(\dfrac{a}{b\left(b+2c\right)^2}\)+\(\dfrac{b}{c\left(c+2a\right)^2}\)+\(\dfrac{c}{a\left(a+2b\right)^2}\)≥\(\dfrac{4}{3}\)
Cho a, b, c là ba số khác nhau, chứng minh rằng:
\(\frac{b-c}{\left(a-b\right)\left(a-c\right)}+\frac{c-a}{\left(b-c\right)\left(b-a\right)}+\frac{a-b}{\left(c-a\right)\left(c-b\right)}=\frac{2}{a-b}+\frac{2}{b-c}+\frac{2}{c-a}\)
cho ba số a,b,c khác nhau:
a)tính \(\frac{ab}{\left(b-c\right)\left(c-a\right)}+\frac{bc}{\left(c-a\right)\left(a-b\right)}+\frac{ca}{\left(a-b\right)\left(b-c\right)}\)
b)chứng minh rằng
\(\frac{a^2}{\left(b-c\right)^2}+\frac{b^2}{\left(c-a\right)^2}+\frac{c^2}{\left(a-b\right)^2}\ge2\)